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How to Solve a System of Linear Equations in Two Variables

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  • 0:04 System of Linear…
  • 1:14 Substitution Method
  • 2:20 Solving the System
  • 2:59 Another Example
  • 4:01 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how you can solve a system of linear equations in two variables by using the substitution method. Learn how easy it is to use on any linear system in two variables.

System of Linear Equations in Two Variables

In this video lesson, you will learn how to solve a system of linear equations in two variables. These are systems of two equations where each equation only has two variables. These variables are usually x and y.

Why should you learn about solving these types of problems? It is because you will come across these types of systems and even larger ones as you progress in math. You will also see that many real-world problems are modeled using these systems. So, being able to solve them easily will be a real help to you.

For example, the number of girls and boys in a classroom can be modeled using a system of linear equations in two variables. If there are a total of 30 students in the classroom and if there are 10 more boys than girls, we can write this system with these two equations:

g + b = 30

g + 10 = b

The first equation tells us that when we add the number of girls, g, and the number of boys, b, together, we get a total of 30 students in the classroom. The second equation tells us that the number of boys is 10 more than the number of girls.

How can we solve this system?

Substitution Method

We can solve it by using the substitution method. This method can be applied to any other system of linear equations in two variables. What this method involves is solving one equation for one of the variables and plugging this into the other equation to solve. Let's see how it works with our system.

We will begin by solving one of the equations for one of the variables. I look closely at my two equations to see which one will be easier to solve for one of the variables, and I see that my second equation is already solved for the variable b, for boys. If this wasn't the case, I can easily solve the first equation for boys by subtracting g from both sides to get b = 30 - g. But, I didn't have to do that because I already have b = g + 10.

Okay. I have what my b equals. I can now substitute this into the other equation. I will substitute b = g + 10 into g + b = 30. I get g + g + 10 = 30. I have just substituted what my b equals into the b variable.

Solving the System

I can now solve for g. I get 2g + 10 = 30. Subtracting 10 from both sides, I get 2g = 20. Dividing by 2 on both sides, I get g = 10. So there are 10 girls.

What about the number of boys? I can find this by substituting the number of girls into the equation for boys that I used in the very beginning. I get b = 10 + 10 = 20. So there are 20 boys.

Is this right? Well there are 10 girls and 20 boys. That makes a total of 30 students. That's right. There are 10 more boys than girls. That is right, too. My answer is correct!

Another Example

Let's try another problem. Let's try solving this system of equations:

2x + y = 10

x - y = -1

We first need to solve one of the equations for one of the variables. I will solve the second equation for x. I add y to both sides to get x = y - 1. Now I substitute this x into the x in the first equation. I get 2(y - 1) + y = 10.

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